4 (a) Sketch the graph of $y = 4 - |2x - 6|$ 4 (b) Solve the inequality 4 - |2x - 6| > 2 - AQA - A-Level Maths Mechanics - Question 4 - 2020 - Paper 1

To sketch the graph of the equation, follow these steps:

1. Identify the Vertex: The expression inside the absolute value, $|2x - 6|$, can be set to zero to find the vertex of the graph:

   $2x - 6 = 0 \Rightarrow x = 3$

   This means the vertex of the absolute value function occurs at $(3, 4)$ because substituting $x = 3$ into the original function gives:

   $y = 4 - |2(3) - 6| = 4 - 0 = 4$

2. Determine Points on Either Side of the Vertex: We can calculate the function values for $x$ values around the vertex. For example:

   • For $x = 2$: $y = 4 - |2(2) - 6| = 4 - |4 - 6| = 4 - 2 = 2$
   • For $x = 4$: $y = 4 - |2(4) - 6| = 4 - |8 - 6| = 4 - 2 = 2$

3. Graph the Points: The graph will be symmetric about the line $x = 3$:

   • Points to plot: $(2, 2)$, $(3, 4)$, $(4, 2)$.

4. Shape of the Graph: The graph forms an inverted 'V' shape with the vertex at $(3, 4)$, intersecting the y-axis at $(0, 4)$. Draw the lines connecting these points to complete the graph.